Conditioning of the weak-constraint variational data assimilation problem for numerical weather prediction

4-Dimensional Variational Data Assimilation (4DVAR) assimilates observations through the minimisation of a least-squares objective function, which is constrained by the model flow. We refer to 4DVAR as strong-constraint 4DVAR (sc4DVAR) in this thesis as it assumes the model is perfect. Relaxing this assumption gives rise to weak-constraint 4DVAR (wc4DVAR), leading to a different minimisation problem with more degrees of freedom. We consider two wc4DVAR formulations in this thesis, the model error formulation and state estimation formulation. The 4DVAR objective function is traditionally solved using gradient-based iterative methods. The principle method used in Numerical Weather Prediction today is the Gauss-Newton approach. This method introduces a linearised `inner-loop' objective function, which upon convergence, updates the solution of the non-linear `outer-loop' objective function. This requires many evaluations of the objective function and its gradient, which emphasises the importance of the Hessian. The eigenvalues and eigenvectors of the Hessian provide insight into the degree of convexity of the objective function, while also indicating the difficulty one may encounter while iterative solving 4DVAR. The condition number of the Hessian is an appropriate measure for the sensitivity of the problem to input data. The condition number can also indicate the rate of convergence and solution accuracy of the minimisation algorithm. This thesis investigates the sensitivity of the solution process minimising both wc4DVAR objective functions to the internal assimilation parameters composing the problem. We gain insight into these sensitivities by bounding the condition number of the Hessians of both objective functions. We also precondition the model error objective function and show improved convergence. We show that both formulations' sensitivities are related to error variance balance, assimilation window length and correlation length-scales using the bounds. We further demonstrate this through numerical experiments on the condition number and data assimilation experiments using linear and non-linear chaotic toy models.

Proper evaluation of the external convective heat transfer for the thermal analysis of cool roofs

Cool materials are characterized by high solar reflectance and high thermal emittance; when applied to the external surface of a roof, they make it possible to limit the amount of solar irradiance absorbed by the roof, and to increase the rate of heat flux emitted by irradiation to the environment, especially during nighttime. However, a roof also releases heat by convection on its external surface; this mechanism is not negligible, and an incorrect evaluation of its entity might introduce significant inaccuracy in the assessment of the thermal performance of a cool roof, in terms of surface temperature and rate of heat flux transferred to the indoors. This issue is particularly relevant in numerical simulations, which are essential in the design stage, therefore it deserves adequate attention. In the present paper, a review of the most common algorithms used for the calculation of the convective heat transfer coefficient due to wind on horizontal building surfaces is presented. Then, with reference to a case study in Italy, the simulated results are compared to the outcomes of a measurement campaign. Hence, the most appropriate algorithms for the convective coefficient are identified, and the errors deriving by an incorrect selection of this coefficient are discussed.